Fluid Flow for the Practicing Chemical Engineer

By James Patrick Abulencia and Louis Theodore

This book teaches the fundamentals of fluid flow by including both theory and the applications of fluid flow in chemical engineering. It puts fluid flow in the context of other transport phenomena such as mass transfer and heat transfer, while covering the basics, from elementary flow mechanics to the law of conservation. The book then examines the applications of fluid flow, from laminar flow to filtration and ventilization. It closes with a discussion of special topics related to fluid flow, including environmental concerns and the economic reality of fluid flow applications.

• Language: English
• Publisher: Wiley
• Release date: Dec 6, 2011
• ISBN: 9781118215715

Book preview

PART I

INTRODUCTION TO FLUID FLOW

This first part of the book provides an introduction to fluid flow. It contains six chapters and each serves a unique purpose in an attempt to treat important introductory aspects of fluid flow. From a practical point-of-view, systems and plants move liquids and gases from one point to another; hence, the student and/or practicing engineer is concerned with several key topics in this area. These receive some measure of treatment in the six chapters contained in this part. A brief discussion of each chapter follows.

Chapter 1 provides an overview of the History of Chemical Engineering—Fluid Flow. Chapter 2 is concerned with Units and Dimensional Analysis. Chapter 3 introduces Key Terms and Definitions. Chapter 4 provides a discussion of Transport Phenomena versus Unit Operations. The final two chapters introduce the reader to Newtonian Fluids (Chapter 5) and Non-Newtonian Flow (Chapter 6). These subjects are important in developing an understanding of the various fluid flow equipment and operations plus their design, which is discussed later in the text.

CHAPTER 1

HISTORY OF CHEMICAL ENGINEERING—FLUID FLOW

1.1 INTRODUCTION

Although the chemical engineering profession is usually thought to have originated shortly before 1900, many of the processes associated with this discipline were developed in antiquity. For example, filtration operations (see Chapter 27) were carried out 5000 years ago by the Egyptians. During this period, chemical engineering evolved from a mixture of craft, mysticism, incorrect theories, and empirical guesses.

In a very real sense, the chemical industry dates back to prehistoric times when people first attempted to control and modify their environment. The chemical industry developed as any other trade or craft. With little knowledge of chemical science and no means of chemical analysis, the earliest chemical engineers had to rely on previous art and superstition. As one would imagine, progress was slow. This changed with time. The chemical industry in the world today is a sprawling complex of raw-material sources, manufacturing plants, and distribution facilities which supplies society with thousands of chemical products, most of which were unknown over a century ago. In the latter half of the nineteenth century, an increased demand arose for engineers trained in the fundamentals of chemical processes. This demand was ultimately met by chemical engineers.

1.2 FLUID FLOW

With respect to fluid flow, the history of pipes and fittings dates back to the Roman Empire. The ingenious engineers of that time came up with a solution for supplying the never-ending demand for fresh water to a city and then disposing of the wastewater produced by the Romans. Their system was based on pipes made out of wood and stone and the driving force of the water was gravity.(1) Over time, many improvements have been made to the piping system. These improvements have included the material choice, shape and size of the pipes; pipes are now made from different metals, plastic, and even glass, with different diameters and wall thicknesses. The next challenge was the connection of the pipes and that was accomplished with fittings. Changes in piping design ultimately resulted from the evolving industrial demands for specific requirements and the properties of fluids that needed to be transported.

The first pump can be traced back to 3000 B.C. in Mesopotamia. It was used to supply water to the crops in the Nile River valley.(2) The pump was a long lever with a weight on one side and a bucket on the other. The use of this first pump became popular in the Middle East and this technology was used for the next 2000 years. Sometimes, a series of pumps would be put in place to provide a constant flow of water to the crops far from the source. Another ancient pump was the bucket chain, a continuous loop of buckets that passed over a pulley-wheel; it is believed that this pump was used to irrigate the Hanging Gardens of Babylon around 600 B.C.(2) The most famous of these early pumps is the Archimedean screw. The pump was invented by the famous Greek mathematician and inventor Archimedes (287-212 B.C.). The pump was made of a metal pipe in which a helix-shaped screw was used to draw water upward as the screw turned. Modern force pumps were adapted from an ancient pump that featured a cylinder with a piston at the top that create[d] a vacuum and [drew] water upward.(2) The first force pump was designed by Ctesibus of Alexandria, Egypt. Leonardo Da Vinci (1452-1519) was the first to come up with the idea of lifting water by means of centrifugal force; however, the operation of the centrifugal pump was first described scientifically by the French physicist Denis Papin (1647-1714) in 1687.(3) In 1754, Leonhard Euler further developed the principles on which centrifugal pumps operate and today the ideal pump performance term, Euler head, is named after him.(4) In the United States, the first centrifugal pump to be manufactured was by the Massachusetts Pump Factory. James Stuart built the first multi-stage centrifugal pump in 1849.(3)

1.3 CHEMICAL ENGINEERING

The first attempt to organize the principles of chemical processing and to clarify the professional area of chemical engineering was made in England by George E. Davis. In 1880, he organized a Society of Chemical Engineers and gave a series of lectures in 1887, which were later expanded and published in 1901 as A Handbook of Chemical Engineering. In 1888, the first course in chemical engineering in the

United States was organized at the Massachusetts Institute of Technology by Lewis M. Norton, a professor of industrial chemistry. The course applied aspects of chemistry and mechanical engineering to chemical processes.(5)

Chemical engineering began to gain professional acceptance in the early years of the twentieth century. The American Chemical Society was founded in 1876 and, in 1908, it organized a Division of Industrial Chemists and Chemical Engineers while authorizing the publication of the Journal of Industrial and Engineering Chemistry. Also in 1908, a group of prominent chemical engineers met in Philadelphia and founded the American Institute of Chemical Engineers.(5)

The mold for what is now called chemical engineering was fashioned at the 1922 meeting of the American Institute of Chemical Engineers when A. D. Little’s committee presented its report on chemical engineering education. The 1922 meeting marked the official endorsement of the unit operations concept and saw the approval of a declaration of independence for the profession.(5) A key component of this report included the following:

Any chemical process, on whatever scale conducted, may be resolved into a coordinated series of what may be termed ‘unit operations,’ as pulverizing, mixing, heating, roasting, absorbing, precipitation, crystallizing, filtering, dissolving, and so on. The number of these basic unit operations is not very large and relatively few of them are involved in any particular process … An ability to cope broadly and adequately with the demands of this (the chemical engineer’s) profession can be attained only through the analysis of processes into the unit actions as they are carried out on the commercial scale under the conditions imposed by practice.

The key unit operations were ultimately reduced to three: Fluid Flow (the subject title of this text), Heat Transfer, and Mass Transfer. The Little report also went on to state that:

Chemical Engineering, as distinguished from the aggregate number of subjects comprised in courses of that name, is not a composite of chemistry and mechanical and civil engineering, but is itself a branch of engineering,…

A time line diagram of the history of chemical engineering between the profession’s founding to the present day is shown in Fig. 1.1.(5) As can be seen from the time line, the profession has reached a crossroads regarding the future education/curriculum for chemical engineers. This is highlighted by the differences of Transport Phenomena and Unit Operations, a topic that is discussed in Chapter 4.

Figure 1.1 Chemical engineering time-line.

REFERENCES

11. http://www.unrv.com/culture/roman-aqueducts.php, 2004.

2. http://www.bookrags.com/sciences/sciencehistory/water-pump-woi.html.

3. A. H. Church and J. Lal, Centrifugal Pumps and Blowers, John Wiley & Sons Inc., Hoboken, NJ, 1973.

4. R. D. Flack, Fundamentals of Jet Propulsion with Applications, Cambridge University Press, New York, 2005.

5. N. Serino, 2005 Chemical Engineering 125th Year Anniversary Calendar, term project, submitted to L. Theodore, 2004.

CHAPTER 2

UNITS AND DIMENSIONAL ANALYSIS

2.1 INTRODUCTION

This chapter is primarily concerned with units. The units used in the text are consistent with those adopted by the engineering profession in the United States. One usually refers to them as the English or engineering units. Since engineers are often concerned with units and conversion of units, both the English and SI system of units are used throughout the book. All the quantities and the physical and chemical properties are expressed using these two systems.

2.1.1 Units and Dimensional Consistency

Equations are generally dimensional and involve several terms. For the equality to hold, each term in the equation must have the same dimensions (i.e., the equation must be dimensionally homogeneous or consistent). This condition can be easily proved. Throughout the text, great care is exercised in maintaining the dimensional formulas of all terms and the dimensional consistency of each equation. The approach employed will often develop equations and terms in equations by first examining each in specific units (feet rather than length), primarily for the English system. Hopefully, this approach will aid the reader and will attach more physical significance to each term and equation.

Consider now the example of calculating the perimeter, P, of a rectangle with length, L, and height, H. Mathematically, this may be expressed as P = 2L + 2H. This is about as simple as a mathematical equation can be. However, it only applies when P, L, and H are expressed in the same units.

A conversion constant/factor is a term that is used to obtain units in a more convenient form. All conversion constants have magnitude and units in the term, but can also be shown to be equal to 1.0 (unity) with no units. An often used conversion constant is

This term is obtained from the following defining equation:

If both sides of this equation are divided by 1 ft one obtains

Note that this conversion constant, like all others, is also equal to unity without any units. Another defining equation is

If this equation is divided by lbf, one obtains

This serves to define the conversion constant gc. Other conversion constants are given in Table A.1 of the Appendix.

Illustrative Example 2.1 Convert the following:

1. 8.03 yr to seconds (s)

2. 150 mile/h to yard/h

3. 100.0 m/s² to ft/min²

4. 0.03 g/cm³ to lb/ft³

Solution

1. The following conversion factors are needed:

365 day/yr

24h/day

60min/h

60s/min

The following is obtained by arranging the conversion factors so that units cancel to leave only the desired units.

2. In a similar fashion,

Terms in equations must also be constructed from a magnitude viewpoint. Differential terms cannot be equated with finite or integral terms. Care should also be exercised in solving differential equations. In order to solve differential equations to obtain a description of the pressure, temperature, composition, etc., of a system, it is necessary to specify boundary and/or initial conditions for the system. This information arises from a description of the problem or the physical situation. The number of boundary conditions (BC) that must be specified is the sum of the highest-order derivative for each independent differential term. A value of the solution on the boundary of the system is one type of boundary condition. The number of initial conditions (IC) that must be specified is the highest-order time derivative appearing in the differential equation. The value for the solution at time equal to zero constitutes an initial condition. For example, the equation

(2.1) equation

requires 2 BCs (in terms of z). The equation

(2.2) equation

requires 1 IC. And finally, the equation

(2.3) equation

requires 1 IC and 2 BCs (in terms of y).

2.2 DIMENSIONAL ANALYSIS

Problems are frequently encountered in fluid flow and other engineering work that involve several variables. Engineers are generally interested in developing functional relationships (equations) between these variables. When these variables can be grouped together in such a manner that they can be used to predict the performance of similar pieces of equipment, independent of the scale or size of the operations, something very valuable has been accomplished.

Consider, for example, the problem of establishing a method of calculating the power requirements for mixing liquids in open tanks. The obvious variables would be the depth of liquid in the tank, the density and viscosity of the liquid, the speed of the agitator, the geometry of the agitator, and the diameter of the tank. There are therefore six variables that affect the power, or a total of seven terms that must be considered. To generate a general equation to describe power variation with these variables, a series of tanks having different diameters would have to be set up in order to gather data for various values of each variable. Assuming that ten different values of each of six variables were imposed on the process, 10⁶ runs would be required. Obviously, a mathematical method for handling several variables that requires considerably less than one million runs to establish a design method must be available. In fact, such a method is available and it is defined as dimensional analysis.(1)

Dimensional analysis is a powerful tool that is employed in planning experiments, presenting data compactly, and making practical predictions from models without detailed mathematical analysis. The first step in an analysis of this nature is to write down the units of each variable. The end result of a dimensional analysis is a list of pertinent dimensionless numbers. A partial list of common dimensionless numbers used in fluid flow analyses is given in Table 2.1.

Table 2.1 Dimensionless numbers

Dimensional analysis is a relatively compact technique for reducing the number and the complexity of the variables affecting a given phenomenon, process or calculation. It can help obtain not only the most out of experimental data but also scale-up data from a model to a prototype. To do this, one must achieve similarity between the prototype and the model. This similarity may be achieved through dimensional analysis by determining the important dimensionless numbers, and then designing the model and prototype such that the important dimensionless numbers are the same in both.

There are three steps in dimensional analysis. These are:

1. List all parameters and their primary units.

2. Formulate dimensionless numbers (or ratios).

3. Develop the relation between the dimensionless numbers experimentally.

Further details on this approach are provided in the next section.

2.3 BUCKINGHAM Pi (π) THEOREM

This theorem provides a simple method to obtain dimensionless numbers (or ratios) termed π parameters. The steps employed in obtaining the dimensionless π parameters are given below(2):

1. List all parameters. Define the number of parameters as n.

2. Select a set of primary dimensions, e.g., kg, m, s, K (English units may also be employed). Let r = the number of primary dimensions.

3. List the units of all parameters in terms of the primary dimensions, e.g., L [=] m, where [=] means has the units of. This is a critical step and often requires some creativity and ingenuity on the part of the individual performing the analysis.

4. Select a number of variables from the list of parameters (equal to r). These are called repeating variables. The selected repeating parameters must include all r independent primary dimensions. The remaining parameters are called nonrepeating variables.

5. Set up dimensional equations by combing the repeating parameters with each of the other non-repeating parameters in turn to form the dimensionless parameters, π. There will be (n — r) dimensionless groups of (πs).

6. Check that each resulting π group is in fact dimensionless.

Note that it is permissible to form a different π group from the product or division of other πs, e.g.,

(2.4) equation

Note, however, that a dimensional analysis approach will fail if the fundamental variables are not correctly chosen. The Buckingham Pi theorem approach to dimensionless numbers is given in the Illustrative Example that follows.

Illustrative Example 2.2 When a fluid flows through a horizontal circular pipe, it undergoes a pressure drop, DLP = (P2 — P1). For a rough pipe, DLP will be higher than a smooth pipe. The extent of non-smoothness of a material is expressed in terms of the roughness, k. For steady state incompressible Newtonian (see Chapter 5) fluid flow, the pressure drop is believed to be a function of the fluid average velocity v, viscosity μ, density p, pipe diameter D, length L, and roughness k (discussed in more detail in Chapter 14), and the speed of sound in fluid (an important variable if the flow is compressible) c, i.e.,

Determine the dimensionless numbers of importance for this flow system.

Solution A pictorial representation of the system in question is provided in Fig. 2.1.

Figure 2.1 Pipe.

List all parameters and find the value of n:

Therefore n = 8.

Choose primary units (employ SI)

List the primary units of each parameter:

Therefore r = 3 with primary units m, s, kg.

Select three parameters from the list of eight parameters. These are the repeating variables:

The non-repeating parameters are then DLP, μ, k, c, and L.

Determine the number of πs:

Formulate the first π, π1, employing DLP as the non-repeating parameter

Determine a, b, and f by comparing the units on both sides of the following equation:

Compare kg:

Compare s:

Compare m:

Substituting back into π1 leads to:

This represents the Euler number (see Table 2.1). Formulate the second π, π2 as

Determine a, b, and f by comparing the units on both sides:

Compare kg:

Compare s:

Compare m:

Substituting back into π2 yields:

Replace π2 by its reciprocal:

where Re = Reynolds number (see Chapter 12).

Similarly, the remaining non-repeating variables lead to

and

Similarly,

Combine the πs into an equation, expressing π1 as a function of π2, π3, π4, and π5:

Consider the case of incompressible flow

The result indicates that to achieve similarity between a model (m) and a prototype (p), one must have the following:

Since Eu =f(Re, k/D, L/D), then it follows that Eum = Eup (see Table 2.1).

2.4 SCALE-UP AND SIMILARITY

To scale-up (or scale-down) a process, it is necessary to establish geometric and dynamic similarities between the model and the prototype. These two similarities are discussed below.

Geometric similarity implies using the same geometry of equipment. A circular pipe prototype should be modeled by a tube in the model. Geometric similarity establishes the scale of the model/prototype design. A 1/10th scale model means that the characteristic dimension of the model is 1/10th that of the prototype.

Dynamic similarity implies that the important dimensionless numbers must be the same in the model and the prototype. For a particle settling in a fluid, it has been shown (see Chapter 23) that the drag coefficient, CD, is a function of the dimensionless Reynolds number, Re, i.e.:

(2.5) equation

By selecting the operating conditions such that Re in the model equals the Re in the prototype, then the drag coefficient (or friction factor) in the prototype equals the friction factor in the model.

REFERENCES

1. I. Farag and J. Reynolds, Fluid Flow, A Theodore Tutorial, East Williston, NY, 1995.

2. W. Badger and J. Banchero, Introduction to Chemical Engineering, McGraw-Hill, New York, 1955.

NOTE: Additional problems are available for all readers at www.wiley.com. Follow links for this title.

CHAPTER 3

KEY TERMS AND DEFINITIONS

3.1 INTRODUCTION

This chapter is concerned with key terms and definitions in fluid flow. Since fluid flow is an important subject that finds wide application in engineering, the understanding of fluid flow jargon is therefore important to the practicing engineer. The handling and flow of either gases or liquids is much simpler, cheaper, and less troublesome than solids. Consequently, the engineer attempts to transport most quantities in the form of gases or liquids whenever possible. It is important to note that throughout this book, the word fluid will always be used to include both liquids and gases.

The mechanics of fluids are treated in most physics courses and form the basis of the subject of fluid flow and hydraulics. Key terms in these two topics that are of special interest to engineers are covered in this chapter. Fluid mechanics includes two topics: statics and dynamics. Fluid statics treats fluids at rest while fluid dynamics treats fluids in motion. The definition of key terms in this subject area is presented in Section 3.2.

3.1.1 Fluids

For the purpose of this text, a fluid may be defined as a substance that does not permanently resist distortion. An attempt to change the shape of a mass of fluid will result in layers of fluid sliding over one another until a new shape is attained. During the change in shape, shear stresses (forces parallel to a surface) will result, the magnitude of which depends upon the viscosity (to be discussed shortly) of the fluid and the rate of sliding. However, when a final shape is reached, all shear stresses will have disappeared. Thus, a fluid at equilibrium is free from shear stresses. This definition applies for both liquids and gases.

3.2 DEFINITIONS

Standard key definitions, particularly as they apply to fluid flow, follow.

3.2.1 Temperature

Whether in a gaseous, liquid, or solid state, all molecules possess some degree of kinetic energy; that is, they are in constant motion—vibrating, rotating, or translating. The kinetic energies of individual molecules cannot be measured, but the combined effect of these energies in a very large number of molecules can. This measurable quantity is known as temperature; it is a macroscopic concept only and as such does not exist on the molecular level.

Temperature can be measured in many ways; the most common method makes use of the expansion of mercury (usually encased inside a glass capillary tube) with increasing temperature. (However, thermocouples or thermistors are more commonly employed in industry.) The two most commonly used temperature scales are the Celsius (or Centigrade) and Fahrenheit scales. The Celsius scale is based on the boiling and freezing points of water at 1-atm pressure; to the former, a value of 100°C is assigned, and to the latter, a value of 0°C. On the older Fahrenheit scale, these temperatures correspond to 212°F and 32°F, respectively. Equations (3.1) and (3.2) show the conversion from one scale to the other:

(3.1) equation

(3.2) equation

where °F = a temperature on the Fahrenheit scale and °C = a temperature on the Celsius scale.

Experiments with gases at low-to-moderate pressures (up to a few atmospheres) have shown that, if the pressure is kept constant, the volume of a gas and its temperature are linearly related (see Chapter 11—Charles’ law) and that a decrease of 0.3663% or (1/273) of the initial volume is experienced for every temperature drop of 1°C. These experiments were not extended to very low temperatures, but if the linear relationship were extrapolated, the volume of the gas would theoretically be zero at a temperature of approximately — 273°C or — 460°F. This temperature has become known as absolute zero and is the basis for the definition of two absolute temperature scales. (An absolute scale is one that does not allow negative quantities.) These absolute temperature scales are the Kelvin (K) and Rankine (°R) scales; the former is defined by shifting the Celsius scale by 273°C so that OK is equal to — 273°C. The Rankine scale is defined by shifting the Fahrenheit scale by 460°.

Equation (3.3) shows this relationship for both absolute temperatures:

(3.3) equation

3.2.2 Pressure

There are a number of different methods used to express a pressure term or measurement. Some of them are based on a force per unit area (e.g., pound-force per square inch, dyne, and so on) and others are based on fluid height (e.g., inches of water, millimeters of mercury, etc.). Pressure units based on fluid height are convenient when the pressure is indicated by a difference between two levels of a liquid. Standard barometric (or atmospheric) pressure is 1 atm and is equivalent to 14.7 psi, 33.91 ft of water, and 29.92 inches of mercury.

Gauge pressure is the pressure relative to the surrounding (or atmospheric) pressure and it is related to the absolute pressure by the following equation:

(3.4) equation

where P is the absolute pressure (psia), Pa is the atmospheric pressure (psi) and Pg is the gauge pressure. The absolute pressure scale is absolute in the same sense that the absolute temperature scale is absolute; i.e., a pressure of zero psia is the lowest possible pressure theoretically achievable—a perfect vacuum.

In stationary fluids subjected to a gravitational field, the hydrostatic pressure difference between two locations A and B is defined as

(3.5) equation

where z is a vertical upwards direction, g is the gravitational acceleration, and p is the fluid density. This equation will be revisited in Chapter 10.

Expressed in various units, the standard atmosphere is equal to 1.00 atmosphere (atm), 33.91 feet of water (ft H2O), 14.7 pound-force per square inch absolute (psia), 2116 pound-force per square foot (psfa), 29.92 inches of mercury (in Hg), 760.0 millimeters of mercury (mm Hg), and 1.013 x 10⁵ Newtons per square meter (N/m²). The pressure term will be reviewed again in several later chapters.

Vapor pressure, usually denoted p’, is an important property of liquids and, to a much lesser extent, of solids. If a liquid is allowed to evaporate in a confined space, the pressure in the vapor space increases as the amount of vapor increases. If there is sufficient liquid present, a point is eventually reached at which the pressure in the vapor space is exactly equal to the pressure exerted by the liquid at its own surface. At this point, a dynamic equilibrium exists in which vaporization and condensation take place at equal rates and the pressure in the vapor space remains constant. The pressure exerted at equilibrium is called the vapor pressure of the liquid. The magnitude of this pressure for a given liquid depends on the temperature, but not on the amount of liquid present. Solids, like liquids, also exert a vapor pressure. Evaporation of solids (called sublimation) is noticeable only for those with appreciable vapor pressures.

3.2.3 Density

At a given temperature and pressure, a fluid possesses density, p, which is measured as mass per unit volume. The density of a fluid depends on both temperature and pressure; if a fluid is not affected by changes in pressure, it is said to be incompressible, and most liquids are incompressible. The density of a liquid can, however, change if there are extreme changes in temperature, and not appreciably affected by moderate changes in pressure. In the case of gases, the density may be affected appreciably by both temperature and pressure. Gases subjected to small changes in pressure and temperature vary so little in density that they can be considered incompressible and the change in density can be neglected without serious error. Density, specific gravity, and other similar properties have the same significance for fluids as for solids.

3.2.4 Viscosity

Viscosity, μ, is an important fluid property that provides a measure of the resistance to flow. The viscosity is frequently referred to as the absolute or dynamic viscosity. The principal reason for the difference in the flow characteristics of water and of molasses is that molasses has a much higher viscosity than water. Note also that the viscosity of a liquid decreases with increasing temperature, while the viscosity of a gas increases with increasing temperature.

One set of units of viscosity in SI units is g/(cm · s), which is defined as a poise (P). Since this numerical unit is somewhat high for many engineering applications, viscosities are frequently reported in centipoises (cP) where one poise is 100 centipoises. In English or engineering units, the dimensions of viscosity are in lb/ft·s. To convert from poises to this unit, one may simply multiply by (30.48/453.6) or (0.0672); to convert from centipoises, multiply by 6.72 x 10−4. To convert centipoises to lb/ft·hr, multiply by 2.42.

Kinematic viscosity, v, is the absolute viscosity divided by the density (μ/p) and has the dimensions of (volume)/length·time. The corresponding unit to the poise is the stoke, having the SI dimensions of cm²/s. The specific viscosity is the ratio of the viscosity to the viscosity of a standard fluid expressed in the same units and measured at the same temperature and pressure. Although all real fluids possess viscosity, an ideal fluid is a hypothetical fluid that has a viscosity of zero and possesses no resistance to shear.

The viscosity is a fluid property listed in many engineering books, including Perry’s Handbook.(1) Data are given as tables, charts, or nomographs. Figures B.1 and B.2 (see Appendix) are two nomographs that can be used to obtain the absolute (or dynamic) viscosity of liquids and gases, respectively.(2,3) In addition, the kinematic viscosities of some common liquids and gases at a temperature of 20°C are listed(2,3) in Tables A.2 and A.3, respectively (see Appendix).

Illustrative Example 3.1 To illustrate the use of nomograph, calculate the dynamic viscosity of a 98% sulfuric acid solution at 45°C.

Solution From Fig. B.1 in the Appendix, the coordinates of 98% H2SO4 are given as X = 7.0 and Y = 24.8 (number 97). Locate these coordinates on the grid and call it point A. From 45°C, draw a straight line through point A and extend it to cut the viscosity axis. The intersection occurs at approximately 12 centipoise (cP). Therefore,

3.2.5 Surface Tension: Capillary Rise

A liquid forms an interface with another fluid. At the surface, the molecules are more densely packed than those within the fluid. This results in surface tension effects and interfacial phenomena. The surface tension coefficient, σ, is the force per unit length of the circumference of the interface, or the energy per unit area of the interface area. The surface tension for water is listed in Table A.4 (see Appendix).

Surface tension causes a contact angle to appear when a liquid interface is in contact with a solid surface, as shown in Fig. 3.1. If the contact angle θ is <90°, the liquid is termed wetting. If θ > 90°, it is a nonwetting liquid. Surface tension causes a fluid interface to rise (or fall) in a capillary tube. The capillary rise is obtained by equating the vertical component of the surface tension force, Fσ, to the weight of the liquid of height h, Fg (see Fig. 3.2). These two forces are shown

Figure 3.1 Surface tension figure.

Figure 3.2 Capillary rise in a circular tube.

in the following equations:

(3.6) equation

(3.7) equation

Equating the above two forces gives:

(3.8) equation

where σ is the surface tension (N/m), θ the contact angle, p the liquid density (kg/m³), g is the acceleration due to gravity (9.807 m/s²), and R is the tube radius (m).

For a droplet, the pressure is higher on the inside than on the outside. The pressure increase in the interior of the liquid droplet is balanced by the surface tension force. By applying a force balance on the interior of a spherical droplet, see Fig. 3.3, one can obtain the force due to the pressure increase, FP, which equals the surface tension force on the ring, FΣ (see Eqs. 3.9 and 3.10). This force balance neglects the weight of the liquid in the droplet

Figure 3.3 Surface tension in a spherical droplet.

(3.9) equation

(3.10) equation

Equating the two forces gives,

(3.11) equation

The pressure increase is therefore,

(3.12) equation

where DLP is the pressure increase (Pa or psi) and r is the droplet radius (m or ft).

Illustrative Example 3.2 A capillary tube is inserted into a liquid. Determine the rise, h, of the liquid interface inside the capillary tube. Data are provided below.

Liquid-gas system is water-air

Temperature is 30°C and pressure is 1 atm

Capillary tube diameter = 8 mm = 0.008 m

Water density = 1000 kg/m³

Contact angle, θ = 0°

Solution The height equation is first written

(3.8) equation

The surface tension of water (see Table A.4 in the Appendix) at 30°C is

The height is therefore

Note that for most industrial applications involving pipes, the diameters are large enough that any capillary rise may be neglected.

Illustrative Example 3.3 At 30°C, what diameter glass tube is necessary to keep the capillary height change of water less than one millimeter? Assume negligible angle of contact.

Solution For air-water-glass, assume the contact angle θ = 0, noting that cos(0°) = 1. Obtain the properties of water from Table A.2 in the Appendix.

Use the capillary rise Equation (3.8) to calculate the tube radius

If the tube diameter is greater than 29 mm, then the capillary rise will be less than 1 mm.

3.2.6 Newton’s Law

The relationship between force mass, velocity, and acceleration may be expressed by Newton’s second law with force equaling the time rate of change of momentum, M.

(3.13)

equation

If the mass is constant,

(3.14) equation

where a = acceleration or dv/dt.

In the English engineering system of units, the pound-force (lbf) is defined as that force which accelerates 1 pound-mass (lb) 32.174 ft/s. Newton’s law must therefore include a dimensional conversion constant for consistency. This constant, gc, is 32.174 (lb/lbf)(ft/s²). When employing SI units, the value of gc becomes unity and has no dimensions associated with it, i.e., gc= 1.0 (see previous chapter for more details). Thus, the gc term is normally retained in equations involving force where English units are employed. The SI unit of force is the Newton (N), which simply expresses force F as the product of mass m and acceleration a (see Equation 3.14 once again). The Newton is defined as the force, when applied to a mass of 1 kg, produces an acceleration of 1 m/s²; the term gc is not retained in this (and similar) equations when SI units are employed.

The term gc is carried in most of the force and force-related terms and equations presented in this and the following chapters. Although both sets of units are employed in the Illustrative Examples and Problems, the reader should note that despite statements to the contrary by academics and theorists, English units are almost exclusively employed by industry in the US.

As described earlier, pressure is a force per unit area. The conversion of force per unit area (S) to a height of fluid follows from Newton’s law, i.e.,

(3.15) equation

and

(3.16) equation

Thus, a vertical column of a given fluid under the influence of gravity exerts a pressure at its base that is directly proportional to its height so that pressure may also be expressed as the equivalent height of a fluid column. The pressure to which a fluid height corresponds may be determined from the density of the fluid and the local acceleration of gravity.

Forces that act on a fluid can be classified as either body forces or surface forces. Body forces are distributed throughout the material, e.g., gravitational, centrifugal, and electromagnetic forces. Body forces therefore act on the bulk of the object from a distance and are proportional to its mass; the most common examples are the aforementioned gravitational and electromagnetic forces. Surface forces are forces that act on the surface of a material. Surface forces are exerted on the surface of the object by other objects in contact with it; they generally increase with increasing contact area. Stress is a force per unit area. If the force is parallel to the surface, the force per unit area is called shear stress. When the force is perpendicular (normal) to a surface, the force per unit area is called normal stress or pressure.

For a stationary (static, non-moving) fluid, the sum of all forces acting on the fluid (ΣF)is zero. Newton’s second law simplifies to

(3.17) equation

When there are two opposing forces, for example, a gravity force and a pressure force, P, (acting on a surface) is then

Equating the two forces gives the result described in Equation (3.15)

(3.18) equation

Illustrative Example 3.4 Given a force F = 10 lbf, acting on a surface of area S = 2 ft², at an angle θ = 30° to the normal of the surface. Determine the magnitude of the normal and parallel force components, the shear stress, and the pressure.

Solution When a force acts at an angle to a surface, the component of the force parallel to that surface is F cos θ. Noting that cos(30°) = 0.866.

The normal (perpendicular) component of the force is F sin θ, noting that sin(30°) = 0.500.

The shear stress, t, is defined as

Likewise, the pressure, P, is defined as

3.2.7 Kinetic Energy

Consider a body of mass, m, that is acted upon by a force, F. If the mass is displaced a distance,